91Ó°ÊÓ

The gradient is a powerful concept from vector calculus. It is a vector that points in the direction of the greatest increase of a function. For a function in two variables, like the given example \(f(x, y) = x^2 - y^2\), the gradient \(abla f\) is calculated using partial derivatives. The formula is:
\[abla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle\]
Partial derivatives measure how a function changes as each variable is varied independently. In our example:

• \(\frac{\partial f}{\partial x} = 2x\)
• \(\frac{\partial f}{\partial y} = -2y\)

Thus, the gradient becomes \(abla f = \langle 2x, -2y \rangle\).
The gradient evaluated at point \(P(1, 0)\) gives \(abla f(1,0) = \langle 2, 0 \rangle\). This vector shows the direction where the function increases fastest and its magnitude tells how fast the increase happens.

A unit vector is a vector that has a length or magnitude of 1. It is primarily used to indicate direction. In the context of the directional derivative problem, we need a direction to obtain the rate of change of our function.
If you have a vector \(\mathbf{v} = \langle a, b \rangle\), its magnitude is \(\sqrt{a^2 + b^2}\). To convert it into a unit vector, divide each component by the magnitude:
\[\mathbf{u} = \left\langle \frac{a}{\sqrt{a^2 + b^2}}, \frac{b}{\sqrt{a^2 + b^2}} \right\rangle\]
In our case, the given vector \(\mathbf{u} = \left\langle \frac{\sqrt{3}}{2}, \frac{1}{2} \right\rangle\) is already a unit vector, meaning its length is already 1, so we can directly use it without further normalization.

The dot product is an operation that takes two equal-length sequences of numbers and returns a single number. This operation is essential when calculating the directional derivative of a function.
Given two vectors \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\), their dot product is
\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\]
For our example, we find the dot product of the gradient at point \(P\), \(abla f(1,0) = \langle 2, 0 \rangle\), with the unit vector \(\mathbf{u} = \left\langle \frac{\sqrt{3}}{2}, \frac{1}{2} \right\rangle\):
\[\langle 2, 0 \rangle \cdot \left\langle \frac{\sqrt{3}}{2}, \frac{1}{2} \right\rangle = 2 \cdot \frac{\sqrt{3}}{2} + 0 \cdot \frac{1}{2} = \sqrt{3}\]
The result, \(\sqrt{3}\), represents the rate of change of the function \(f\) at point \(P\) in the direction of \(\mathbf{u}\).

Partial derivatives are fundamental in multivariable calculus. They allow us to understand how a function changes when altering just one of its variables while keeping others constant.
For a function \(f(x, y)\), the partial derivative with respect to \(x\) is written as \(\frac{\partial f}{\partial x}\). It tells us the rate of change of the function as \(x\) changes, fixing \(y\). Similarly, \(\frac{\partial f}{\partial y}\) is the rate of change with respect to \(y\).
In our specific function \(f(x, y) = x^2 - y^2\), the partial derivatives are:

• \(\frac{\partial f}{\partial x} = 2x\)
• \(\frac{\partial f}{\partial y} = -2y\)

These derivatives tell us how the function will increase or decrease as \(x\) or \(y\) is varied, giving insight into the function's behavior in each direction. They are the building blocks for calculating gradients, and hence, directional derivatives.